Given 3 integers, can there be a triangle with edges equal to those lengths?

I have never encountered even the tiniest of geometric questions in my professional career. On the other hand, many people have worked in graphics and games, where geometric reasoning (presumably) matters a lot. As it happens, even my stunted geometric reasoning was able to handle this question. I think it’ll be a nice exercise on a whiteboard as a general sort of problem-solving measurement, but I don’t see how it says much about the interviewee’s actual software-development capabilities.

Ultimately, what the question is asking for is a list of constraints. If those constraints are all satisfied, then there can exist a triangle with edge-lengths a, b, c. If those constraints aren’t satisfied, then there isn’t.

Finding Constraints

So we have to determine those constraints. What are those constraints? The way I think about them is that I imagine the constraints are true for (a, b, c) so I ask myself something like: “if a, b, c from a triangle (so to speak), and a is really big (relative to b and c), what does b and c look like?”

Sketching out that question on paper, it seems that the “larger” a gets relative to b and c, the “flatter” the triangle has to be. Turning it around, b and c together are always going to be a bit bigger than a. So can hypothesize: “a < b + c”, and pretty quickly establish analogous constraints with the other 2 edges.

So that’s a nice constraint! We can play around with the opposite: say a is 100, and b and c are both 2. It seems pretty clear that b and c won’t be able to touch each other, when placed on either end of a. So that’s reassuring.

Looking at the constraint we have, we have what’s called an “upper bound” on a. We know that the size of a can only get so big before it breaks the rules. Can we go the other way? If we know b and c, does a have to be a certain length? Intuitively, we can imagine a sort of arm or hinge, where one edge is b, the other c. Say c=b/2. Clearly, even if we close the hinge entirely, the end of c is pretty far away from the other end of b. It suggests that the length of a must be at least the difference between c and b, or a > b – c.

Do fun arithmetic, and those are the same constraints as we determined above, just with a sign change. Is this a real proof? Not in my opinion. But at this point I have intuitive arguments that shows the 3 constraints we determined (a<b+c,b<a+c,c<a+b) effectively both upper-and-lower bound the value of each a, b, c, so I got confident enough to hit the go button. Hey, it passed all the tests.

Here are those constraints, plus an edge case handler, written down:

def is_triangle(a, b, c):
    if a < 0 or b < 0 or c < 0:
        return False
    b1 = a + b > c
    b2 = a + c > b
    b3 = b + c > a
    return b1 and b2 and b3

Should I have included pictures explaining my reasoning in this post? Almost certainly yes.

I don’t often explore geometric problems, and I don’t think their a focus for my students, but as more come up the solutions will be found here.